If y = \int (12x^2 + 6\sin(2x)) \, dx and y = 10 when x = 0, what is the value of the cons… — JAMB UTME

If $y = \int (12x^2 + 6\sin(2x)) \, dx$ and $y = 10$ when $x = 0$ , what is the value of the constant of integration $C$ ?

Answer: 13

The value of the constant of integration

C

13

Integrating

12x^2 + 6\sin(2x)

gives

y = 4x^3 - 3\cos(2x) + C

. Substituting

x = 0

yields

y(0) = 4(0)^3 - 3\cos(0) + C = -3 + C

. Setting

-3 + C = 10

and solving for

C

gives

C = 13

Integrate the function with respect to

x

y = 4x^3 - 3\cos(2x) + C

Applying the power rule

\int ax^n \, dx = \frac{ax^{n+1}}{n+1}

and trigonometric integration rule

\int k\sin(bx) \, dx = -\frac{k}{b}\cos(bx)

Apply the initial condition

x = 0

and

y = 10

10 = 4(0)^3 - 3\cos(0) + C

Substituting the boundary values to solve for the specific constant of integration.

Evaluate trigonometric function and solve for

C

C = 13

Since

\cos(0) = 1

, the equation simplifies to

10 = -3 + C

, leading directly to

C = 13

Indefinite Integration of Polynomial and Trigonometric Functions with Initial Conditions

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